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mathematics
calculus with applications

Calculus With Applications 12th Edition Margaret L. Lial - Solutions

In Exercises, determine the interval over which the function is continuous.ƒ(x) = 4x2 - 3x + 7
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = -4x2 + 11x
If the instantaneous rate of change of ƒ(x) with respect to x is positive when x = 1, is f increasing or decreasing there?
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = 6x2 - 4x
Let(a) Graph the function.(b) Determine ƒ(2) and limx→2 ƒ(x). Is ƒ(2) continuous at x = 2?(c) Determine ƒ(3) and limx→3 ƒ(x). Is ƒ(x) continuous at x = 3?(d) Determine ƒ(4) and limx→4 ƒ(x). Is ƒ(x) continuous at x = 4? f(x) = 1 if x < 2 x + 3 if 2 ≤ x ≤ 4. 7 if x > 4
Suppose that the total profit in hundreds of dollars from selling x items is given byP(x) = 2x2 - 5x + 6.Find the average rate of change of profit for the following changes in x.(a) 2 to 4 (b) 2 to 3
The unemployment rates in the United States for the years 2000–2019 are shown in the graph below. Find the average change per year in unemployment for the following time periods.(a) 2000 to 2010 (b) 2010 to 2016(c) 2010 to 2019 Unemployment rate, % 12 10 00 4.0 5.8 6.0 4.7
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = ex
The revenue (in thousands of dollars) from producing x units of an item isR(x) = 10x - 0.002x2.(a) Find the average rate of change of revenue when production is increased from 1000 to 1001 units.(b) Find and interpret the instantaneous rate of change of revenue with respect to the number of items
In Exercises, (a) Graph the given function, (b) Find all values of x where the function is discontinuous, and (c) Find the limit from the left and from the right at any values of x found in part (b). g(x) 0 x²5x 5 if x < 0 if 0≤x≤ 5 if x > 5
In Exercises, (a) Graph the given function, (b) Find all values of x where the function is discontinuous, and (c) Find the limit from the left and from the right at any values of x found in part (b). g(x) = 11 if x < -1 x² + 2 if-1 ≤ x ≤ 3 11 if x > 3
The average price (in cents) per gallon of unleaded gasoline in the United States for the years 2010 to 2019 is shown in the following chart. Find the average rate of change per year in the average price per gallon for each time period.(a) 2010 to 2012 (b) 2012 to 2019(c) 2010 to 2019
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = ln |x|
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = -2/x
If $1000 is invested in an account that pays 3% compounded annually, the total amount, A(t), in the account after t years isA(t) = 100011.032t.(a) Find the average rate of change per year of the total amount in the account for the first five years of the investment (from t = 0 to t = 5).(b) Find
Find the x-values where the following do not have derivatives. VA 2+ 0 -2 +2 X
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = 6/x
If $1000 is invested in an account that pays 3% compounded continuously, the total amount, A(t), in the account after t years isA(t) = 1000e0.03t.(a) Find the average rate of change per year of the total amount in the account for the first five years of the investment (from t = 0 to t = 5).(b) Find
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim (2x² - 4x + 5) x-3
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = √x
The graph shows the money remaining in the Medicare Trust Fund at the end of the fiscal year.Find the approximate average rate of change in the trust fund for each time period.(a) From 2015 to 2018 (b) From 2018 to 2028 Dollars (in billions) 400 300 200 100 historical estimates: 0 15 16 17 18
Use a graphing calculator to find f′(2), f′(16), and f′(-3) for the following when the derivative exists. (Round to four decimal places, when necessary.)ƒ(x) = -3 √x
In Exercises, (a) Graph the given function, (b) Find all values of x where the function is discontinuous
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim (√x - x) x→8
In Exercises, (a) Graph the given function, (b) Find all values of x where the function is discontinuous, and (c) Find the limit from the left and from the right at any values of x found in part (b). h(x): = x² + x 3 - x 12 if x ≤ 1 if x > 1
The graph shows the temperature T in degrees Fahrenheit as a function of the altitude h in feet when an inversion layer is over Southern California. (An inversion layer is formed when air at a higher altitude, say 3000 ft, is warmer than air at sea level, even though air normally is cooler with
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x-2 9- X-zX x x + 2
The following chart shows how the percentage of eighth graders, tenth graders, and twelfth graders who have used marijuana in their lifetime has varied in recent years.(a) Find the average annual rate of change in the percent of eighth graders who have used marijuana in their lifetime over the
In Exercises, (a) Use a graphing calculator to tell where the rational function P(x)/Q(x) is discontinuous, and (b) Verify your answer from part (a) by using the graphing calculator to plot Q(x) and determine where Q(x) = 0. You will need to choose the viewing window carefully. f(x) = x²
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x-5 x². X - 3x x-5 10
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x-0 1/(x + 3) - 1/3 X
A car is moving along a straight test track. The position in feet of the car, s(t), at various times t is measured, with the following results.Find and interpret the average velocities for the following changes in t.(a) 0 to 2 seconds (b) 2 to 4 seconds(c) 4 to 6 seconds (d) 6 to 8
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 9-x^ lim x→36 x — 36
In Exercises, find the value of the constant k that makes the function continuous. f(x) = if x ≤ 2 Skx² [√x + k if x > 2
The future size of the world population depends on how soon it reaches replacement-level fertility, the point at which each woman bears on average about 2.1 children. The graph shows projections for reaching that point in different years.(a) Estimate the average rate of change in population for
Find the x-values where the following do not have derivatives. + -6 + y 8 0 6 X
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x-2x² x + 2 - x - 6
Find the x-values where the following do not have derivatives. -3 1 1 I 1 1 1 f(x) + 0 -3- X
Find the x-values where the following do not have derivatives. ا f(x) 3 نرا 10 2 | | | | ا | X
In Exercises, find the value of the constant k that makes the function continuous. g(x) = [x³ + k_ifx ≤ 3 kx 5 if x > 3 -
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. x - 3 5x +4 lim x→-3x2 +
In Exercises, find the value of the constant k that makes the function continuous. h(x) = 3x² + 2x x + 2 3x + k 8 if x = -2 if x = -2
In Exercises, find the value of the constant k that makes the function continuous. g(x) = 2x² - x - x-3 kx - 1 15 if x # 3 if x = 3
The number of cases of measles in the United States for recent years is shown in the graph at the top of the next column. Find the average rate of change per year for each of the following time periods.(a) 2010 to 2014(b) 2014 to 2017(c) 2010 to 2019 Measles
The crown length (as shown below) of first molars in fetuses is related to the postconception age of the tooth asL(t) = -0.01t2 + 0.788t - 7.048,where L(t) is the crown length, in millimeters, of the molar t weeks after conception.(a) Find the average rate of growth in crown length during weeks 22
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 3 X E-X 6-zx lim
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. x² lim x 2 x + 4 2
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x→1 5x² - 7x + 2 x² - 1
Find the x-values where the following do not have derivatives.Suppose g(x) = 3√x . Use the graph of g(x) to find g102.
In Exercises, (a) Use a graphing calculator to tell where the rational function P(x)/Q(x) is discontinuous, and (b) Verify your answer from part (a) by using the graphing calculator to plot Q(x) and determine where Q(x) = 0. You will need to choose the viewing window carefully. f(x)
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. x² - 9 2 X lim x→−3 x² + x − 6
Explain in your own words what the Intermediate Value Theorem says and why it seems plausible.
The metabolic rate of a person who has just eaten a meal tends to go up and then, after some time has passed, returns to a resting metabolic rate. This phenomenon is known as the thermic effect of food. Researchers have indicated that the thermic effect of food (in kJ/hr) for a particular person
By considering, but not calculating, the slope of the tangent line, give the derivative of the following.(a) ƒ(x) = 5 (b) ƒ(x) = x(c) ƒ(x) = -x (d) The line x = 3(e) The line y = mx + b
Explain why limx→2 (3x2 + 8x) can be evaluated by substituting x = 2.
The following graph shows how immigration (in thousands) to the United States has varied from 1910 to 2010. The following chart shows how the percentage of eighth graders, tenth graders, and twelfth graders who have used marijuana in their lifetime has varied in recent years.(a) Find the average
If the rate of change of ƒ(x) is zero when x = a, what can be said about the tangent line to the graph of ƒ(x) at x = a?
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x-0 -1/(x + 2) + 1/2
The graph shows the profit from the daily production of x thousand kilograms of an industrial chemical. Use the graph to find the following limits.(a) limx→6 P(x) (b) limx→10- P(x)(c) limx→10+ P(x)(c) limx→10 P(x)(e) Where is the function discontinuous? What might account for
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. √x - 5 lim x 25 x 25
In Exercises, find the derivative of the function at the given point.(a) Approximate the definition of the derivative with small values of h.(b) Use a graphing calculator to zoom in on the function until it appears to be a straight line, and then find the slope of that line.ƒ(x) = xx; a = 2
The cost to transport a mobile home depends on the distance, x, in miles that the home is moved. Let C(x) represent the cost to move a mobile home x miles. One firm charges a flat fee of $100, plus an additional cost per mile calculated as follows.Find the cost to move a mobile home the following
Consider the example at the beginning of this section regarding the car traveling from San Francisco.(a) Estimate the instantaneous velocity at 1 hour. Assume that the velocity changes at a steady rate from one half-hour to the next.(b) Estimate the instantaneous velocity at 2 hours.
In Exercises, find the derivative of the function at the given point.(a) Approximate the definition of the derivative with small values of h.(b) Use a graphing calculator to zoom in on the function until it appears to be a straight line, and then find the slope of that line.ƒ(x) = xx; a = 3
The distance in feet of a particle from some fixed point is given bys(t) = t2 + 5t + 2,where t is time measured in seconds. Find the average velocity of the particle over the following intervals.(a) 4 to 6 seconds(b) 4 to 5 seconds(c) Find the instantaneous velocity of the particle when t = 4.
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim h→0 (x + h)² - x² h
In Exercises, find the derivative of the function at the given point.(a) Approximate the definition of the derivative with small values of h.(b) Use a graphing calculator to zoom in on the function until it appears to be a straight line, and then find the slope of that line.ƒ(x) = x1/x; a = 3
For each function in Column A, graph [ƒ(x + h) - ƒ(x)4/h for a small value of h on the window [-2, 2] by [-2, 8]. Then graph each function in Column B on the same window. Compare the first set of graphs with the second set to choose from Column B the derivative of each of the functions in Column
A company charges $1.25 per lb for a certain fertilizer on all orders 100 lb or less, and $1 per lb for orders over 100 lb. Let F(x) represent the cost for buying x lb of the fertilizer. Find the cost of buying the following.(a) 80 lb (b) 150 lb (c) 100 lb(d) Where is F discontinuous?
Recently, a car rental firm charged $36 per day or portion of a day to rent a car for a period of 1 to 5 days. Days 6 and 7 were then free, while the charge for days 8 through 12 was again $36 per day. Let A(t) represent the average cost to rent the car for t days, where 0 < t ≤ 12. Find the
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim h→0 (x + h)³ = x³ h
In 2020, it cost $1.00 to send a large envelope within the United States for the first ounce, and $0.20 for each additional ounce, or fraction thereof, up to 13 ounces. Let C(x) be the cost to mail x ounces. Find the following.(a) limx→3 C(x)(b) limx→3+ C(x)(c) limx→3C(x)(d) C(3)(e)
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim (x² + 4x 3) - X-∞
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim (x³ + 2x + 8) X--∞
A woman’s weight naturally increases during the course of a pregnancy. When she delivers, her weight immediately decreases by the approximate weight of the child. Suppose that a 120-lb woman gains 27 lb during pregnancy, delivers a 7-lb baby, and then, through diet and exercise, loses the
Suppose the demand for a certain item is given by D(p) = -2p2 - 4p + 300, where p represents the price of the item in dollars.(a) Find the rate of change of demand with respect to price.(b) Find and interpret the rate of change of demand when the price is $10.
Researchers at Iowa State University and the University of Arkansas have developed a piecewise function that can be used to estimate the body weight (in grams) of a male broiler during the first 56 days of life according towhere t is the age of the chicken (in days). (A broiler is a chicken raised
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x→ 7x 3x 1
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 8x + 2 lim x--∞ 4x - 5
The revenue in dollars generated from the sale of x picnic tables is given by(a) Find the marginal revenue.(b) Estimate the revenue from the sale of the 1001st table by finding R′(1000).(c) Determine the actual revenue from the sale of the 1001st table.(d) Compare your answers for parts (b) and
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 3x² + 2x lim x→∞ 2x²² - 2x + 1 -00
The average cost per unit, C̅(x), in dollars, to produce x units of yogurt is given by(a) Find the marginal average cost.(b) Find and interpret the marginal cost at a production level of 50 units of yogurt.(c) Find and interpret the marginal cost at a production level of 100 units of yogurt.
The profit (in thousands of dollars) from the expenditure of x thousand dollars on advertising is given by P(x) = 1000 + 32x - 2x2. Find the marginal profit at the following expenditures. In each case, decide whether the firm should increase the expenditure.(a) $8000 (b) $6000(c)
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim 00-X x² + 2x - 5 3x² + 2
Suppose a cost-benefit model is given bywhere ƒ(x) is the cost (in thousands of dollars) of removing x percent of a certain pollutant from an area stream.(a) Find the rate of change of the cost with respect to the percent of pollution removed.(b) Find and interpret the rate of change of cost when
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 3x³ + 2x - 1 3x³ - 2 lim xx2x4
The cost in dollars of producing x tacos is C(x) = -0.00375x2 + 1.5x + 1000, for 0 ≤ x ≤ 180.(a) Find the marginal cost.(b) Find and interpret the marginal cost at a production level of 100 tacos.(c) Find the exact cost to produce the 101st taco.(d) Compare the answers to parts (b) and (c). How
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 2.x² 1 1 lim x3x² + 2
Many birds, such as cockatiels or the Arctic terns shown below, have flight muscles whose expenditure of power varies with the flight speed in a manner similar to the graph shown on the next page. The horizontal axis of the graph shows flight speed in meters per second, and the vertical axis shows
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 2x³ - x - 3 lim x→∞0 6x² 2 - x - 1 I
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. 2x² - 7x4 lim x 9x2 + 5x - 6
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. -5x³ 4x² + 8 lim x 6x² + 3x + 2
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x →∞ x4 - x³ - 3x 7x² + 9 2
The graph below shows the temperature one Christmas day of a Heat-Kit Bakeoven, a wood-burning oven for baking. The oven was lit at 8:30 a.m. Let T(x) be the temperature x hours after 8:30 a.m.(a) Find and interpret T′(0.5).(b) Find and interpret T′(3).(c) Find and interpret T′(4).(d) At a
In one research study, the population of a certain shellfish in an area at time t was closely approximated by the following graph. Estimate and interpret the derivative at each of the marked points. y 12,000 9000 6000 3000 2 4 6 8 10 12 14 16 18 20
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. Let g(x) = So 1/1/x² – 3 - if x = -2 -2 3 if x Find lim g(x). x--2
In Exercises, use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. Let f(x) = = [x³ + 2 if x # −1 5 if x = -1 Find lim f(x). x-1
The graph shows how the velocity of a baseball that was traveling at 90 miles per hour when it was hit by a Major League baseball player varies with respect to the weight of the bat.(a) Estimate and interpret the derivative for a 40-oz and 30-oz bat.(b) What is the optimal bat weight for this
The graph shows how the velocity of a baseball that was traveling at 40 miles per hour when it was hit by a Little League baseball player varies with respect to the weight of the bat.(a) Estimate and interpret the derivative for a 16-oz and 25-oz bat.(b) What is the optimal bat weight for this
The eating behavior of a typical human during a meal can be described byI(t) = 27 + 72t - 1.5t2,where t is the number of minutes since the meal began, and I(t) represents the amount (in grams) that the person has eaten at time t.(a) Find the rate of change of the intake of food for this particular

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